Natural implicative expansions of variants of Kleene's strong 3-valued logic with Gödel-type and dual Gödel-type negation

Abstract

Let MK3 and MK3 be Kleene's strong 3-valued matrix with only one and two designated values, respectively. Next, let MK3 (resp., MK3 ) be defined exactly as MK3 (resp., MK3 ), except that the characteristic Łukasiewicz-type negation of Kleene's strong 3-valued matrix is replaced by a ‘Gödel-type’ negation (resp., ‘dual Gödel-type’ negation). The aim of this paper is to axiomatize the logics determined by all natural implicative expansions of MK3 and MK3 . The axiomatic formulations are defined by using a ‘two-valued’ Belnap-Dunn semantics.